There is an acute and increasing need to adapt standard statistical methods and to develop new approaches for the analysis of very large data sets. A data set is very large if it raises very difficult or insurmountable computational problems for standard data analysis using available computing systems. The accelerated increase in size and complexity of data sets is due in part to increased computational and storage capabilities, new measurement technologies, study designs, and an increasing number of study units. This proposal is concerned with statistical methods for the analysis of an emerging type of very large data set, where very high dimensional outcomes and predictors, such as images or densely sampled biosignals, are recorded at multiple visits on hundreds or thousands of subjects. The methods proposed will describe the cross-sectional, longitudinal and measurement error variability in longitudinal studies where observed data are functions or images. Methods for scalar on function/image regression analysis will also be addressed for the case of very highly dimensional predictors. The proposed methodology is inspired by and applied to very large studies of sleep and Diffusion Tensor Imaging (DTI) brain tractography.